Roughly speaking, an attractor of a dynamical system is a subset of the state space to which orbits originating from typical initial conditions tend as time increases. It is very common for dynamical systems to have more than one attractor. For each such attractor, its basin of attraction is the set of initial conditions leading to long-time behavior that approaches that attractor. Thus the qualitative behavior of the long-time motion of a given system can be fundamentally different depending on which basin of attraction the initial condition lies in (e.g., attractors can correspond to periodic, quasiperiodic or chaotic behaviors of different types). Regarding a basin of attraction as a region in the state space, it has been found that the basic topological structure of such regions can vary greatly from system to system. In what follows we give examples and discuss several qualitatively different kinds of basins of attraction and their practical implications.

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Example

A simple example is that of a point particle moving in a two-well potential with friction, as in Figure 1(a). Due to the friction, all initial conditions, except those at \( x=dx/dt=0 \) or on its stable manifold eventually come to rest at either \( x=x_0 \) or \( x=-x_0 \ ,\) which are the two attractors of the system. A point initially placed on the unstableequilibrium point, \( x=0, \) will stay there forever; and this state has a one-dimensional stable manifold. Figure 1(b) shows the basins of attraction of the two stable equilibrium point, \( x=\pm x_0, \) where the crosshatched region is the basin for the attractor at \( x=x_0 \) and the blank region is the basin for the attractor at \( x=-x_0 \ .\) The boundary separating these two basins is the stable manifold of the unstable equilibrium \( x=0.\)

Fractal basin boundaries

In the above example, the basin boundary was a smooth curve. However, other possibilities exist. An example of this occurs for the map
\[ x_{n+1}=(3x_n)\mod 1\ ,\]
\[ y_{n+1}=1.5y_n + \cos 2\pi x_n\ .\]
For almost any initial condition (except for those precisely on the boundary between the basins of attraction), \( \lim _{n\rightarrow \infty} y_n \) is either \( y=+\infty \) or \( y=-\infty \ ,\) which we may regard as the two attractors of the system. Figure 2 shows the basin structure for this map, with the basin for the \( y=-\infty \) attractor black and the basin of the \( y=+\infty \) attractor blank. In contrast to the previous example, the basin boundary is no longer a smooth curve. In fact, it is a fractal curve with a box-counting dimension 1.62.... We emphasize that, although fractal, this basin boundary is still a simple curve (it can be written as a continuous parametric functional relationship \( x=x(s), y=y(s) \) for \( 1>s>0 \) such that \( (x(s_1), y(s_1))\neq (x(s_2), y(s_2)) \) if \( s_1\neq s_2\ .\))

Figure 3: Basins of attraction for a forced damped pendulum (picture made by H.E. Nusse).

Another example of a system with a fractal basin boundary is the forced damped pendulum equation,
\[d^2\theta /dt^2+0.1 d\theta /dt+\sin \theta =2.1 \cos t\ .\]
For these parameters, there are two attractors which are both periodic orbits (Grebogi, Ott and Yorke, 1987). Figure 3 shows the basins of attraction of these two attractors with initial \(\theta\) values plotted horizontally and initial values of \(d\theta /dt\) plotted vertically. The figure was made by initializing many initial conditions on a fine rectangular grid. Each initial condition was then integrated forward to see which attractor its orbit approached. If the orbit approached a particular one of the two attractors, a black dot was plotted on the grid. If it approached the other attractor, no dot was plotted. The dots are dense enough that they fill in a solid black region except near the basin boundary. The speckled appearance of much of this figure is a consequence of the intricate, finescaled structure of the basin boundary. In this case the basin boundary is again a fractal set (its box-counting dimension is about 1.8), but its topology is more complicated than that of the basin boundary of Figure 2 in that the Figure 3 basin boundary is not a simple curve. In both of the above examples in which fractal basin boundaries occur, the fractality is a result of chaotic motion (see transient chaos) of orbits on the boundary, and this is generally the case for fractal basin boundaries (McDonald et al., 1985).

Basin Boundary Metamorphoses

We have seen so far that there can be basin boundaries of qualitatively different types. As in the case of attractors, bifurcations can occur in which basin boundaries undergo qualitative changes as a system parameter passes through a critical bifurcation value. For example, for a system parameter \( p<p_c \ ,\) the basin boundary might be a simple smooth curve, while for \( p>p_c \) it might be fractal. Such basin boundary bifurcations have been called metamorphoses (Grebogi, et al., 1987).

The Uncertainty Exponent

Fractal basin boundaries, like those illustrated above, are extremely common and have potentially important practical consequences. In particular, they may make it more difficult to identify the attractor corresponding to a given initial condition, if that initial condition has some uncertainty. This aspect is already implied by the speckled appearance of Figure 3. A quantitative measure of this is provided by the uncertainty exponent (McDonald et al., 1985). For definiteness, suppose we randomly choose an initial condition with uniform probability density in the area of initial condition space corresponding to the plot in Figure 3. Then, with probability one, that initial condition will lie in one of the basins of the two attractors [the basin boundary has zero Lebesgue measure (i.e., 'zero area') and so there is zero probability that a random initial condition is on the boundary]. Now assume that we are also told that the initial condition has some given uncertainty, \( \epsilon \ ,\) and, for the sake of illustration, assume that this uncertainty can be represented by saying that the real initial condition lies within a circle of radius \( \epsilon \) centered at the coordinates \( (x_0,y_0) \) that were randomly chosen. We ask what is the probability that the \( (x_0,y_0) \) could lie in a basin that is different from that of the true initial condition, i.e., what is the probability, \( \rho (\epsilon ) \ ,\) that the uncertainty \( \epsilon \) could cause us to make a mistake in a determination of the attractor that the orbit goes to. Geometrically, this is the same as asking what fraction of the area of Figure 3 is within a distance \( \epsilon \) of the basin boundary. This fraction scales as
\[\rho (\epsilon )\sim \epsilon ^\alpha\ ,\]
where \( \alpha \) is the uncertainty exponent (McDonald et al., 1985) and is given by \( \alpha =D-D_0 \) where \( D \) is the dimension of the initial condition space (\( D=2 \) for Figure 3) and \( D_0 \) is the box-counting dimension of the basin boundary. For the example of Figure 3, since \( D_0\cong 1.8 \ ,\) we have \( \alpha \cong 0.2 \ .\) For small \( \alpha \) it becomes very difficult to improve predictive capacity (i.e., to predict the attractor from the initial condition) by reducing the uncertainty. For example, if \( \alpha =0.2 \ ,\) to reduce \( \rho (\epsilon )\) by a factor of 10, the uncertainty \( \epsilon \) would have to be reduced by a factor of \( 10^5 \ .\) Thus, fractal basin boundaries (analogous to the butterfly effect of chaotic attractors) pose a barrier to prediction, and this barrier is related to the presence of chaos.

Riddled Basins of Attraction

We now discuss a type of basin topology that may occur in certain special systems; namely, systems that, through a symmetry or some other constraint, have a smooth invariant manifold. That is, there exists a smooth surface or hypersurface in the phase space, such that any initial condition in the surface generates an orbit that remains in the surface. These systems can have a particularly bizarre type of basin structure called a riddled basin of attraction (Alexander et al., 1992; Ott et al., 1994). In order to discuss what this means, we first have to clearly state what we mean by an "attractor". For the purposes of this discussion, we use the definition of Milnor (1985): a set in state space is an attractor if it is the limit set of orbits originating from a set of initial conditions of positive Lebesgue measure. That is, if we randomly choose an initial condition with uniform probability density in a suitable sphere of initial condition space, there is a non-zero probability that the orbit from the chosen initial condition goes to the attractor. This definition differs from another common definition of an attractor which requires that there exists some neighborhood of an attractor such that all initial conditions in this neighborhood generate orbits that limit on the attractor. As we shall see, an "attractor" with a riddled basin conforms with the first definition, but not the second definition. The failure to satisfy the second definition is because there are points arbitrarily close to an attractor with a riddled basin, such that these points generate orbits that go to another attractor (hence the neighborhood mentioned above does not exist.)

Figure 4: Schematic illustration of a situation with a riddled basin of attraction.

We are now ready to say what we mean by a riddled basin. Suppose our system has two attractors which we denote \(A\) and \(C\) with basins \( \hat A \) and \( \hat C \ .\) We say that the basin \( \hat A \) is riddled, if, for every point \(p\) in \( \hat A \ ,\) an \( \epsilon \)-radius ball, \( B_\epsilon (p)\) centered at \( p \) contains a positive Lebesgue measure of points in \( \hat C\) for any \( \epsilon >0\ .\) This circumstance has the following surprising implication. Say we initialize a state at \(p\) and find that the resulting orbit goes to \(\hat A\ .\) Now say that we attempt to repeat this experiment. If there is any error in our resetting of the initial condition, we cannot be sure that the orbit will go to \(A\) (rather than \(C\)), and this is the case no matter how small our error is. Put another way, even though the basin \(\hat A\) has positive Lebesgue measure (non-zero volume), the set \(\hat A\) and its boundary set are the same. Thus the existence of riddled basins calls into question the repeatability of experiments in such situations. Figure 4 illustrates the situation we have been discussing. As shown in Figure 4, the attractor with a riddled basin lies on a smooth invariant surface (or manifold) \(S\ ,\) and this is general for attractors with riddled basins. Typical systems do not admit smooth invariant manifolds, and this is why riddled basins (fortunately?) do not occur in generic cases. Examples, where a dynamical system has a smooth invariant surface are a system with reflection symmetry of some coordinate \(x\) about \(x=0\ ,\) in which case \(x=0\) would be an invariant manifold, and a predator-prey model in population dynamics, in which case one of the populations being zero (extinction) is an invariant manifold of the model.